Parameter estimating method based on correction of independence of error term, computer program for implementing the same method, and system configured to perform the same method

ABSTRACT

Disclosed are a parameter estimating method based on correction of independence of an error term, a computer program implementing the method, and a system configured to perform the method. The method includes: acquiring a measured velocity, which is a velocity of a moving object measured with a speedometer being operated by an observer and having a specific measurement unit, and an actual velocity of the moving object with respect to the observer; performing modeling of a linear regression model with the measured velocity as a dependent variable and the actual velocity of the moving object with respect to the observer as an independent variable; and estimating parameters of a linear regression equation after correcting the independence of the error term based on Lorentz velocity transformation.

CROSS-REFERENCE TO PRIOR APPLICATION

This application claims priority to Korean Patent Application No.10-2020-0033460 (filed on Mar. 18, 2020), which is hereby incorporatedby reference in its entirety.

BACKGROUND

The present disclosure relates to a technology of estimating ameasurement unit and accuracy of a speedometer and, more particularly,to a parameter estimating method based on correction of independence ofan error term, a computer program for implementing the method, and asystem configured to perform the method, the method which is capable ofstatistically estimating a measurement unit of a speedometer andaccuracy thereof, the speedometer having an error and measures avelocity of an object in an unknown measurement unit under the specialtheory of relativity.

In statistics, linear regression is a regression analysis approach tomodeling the linear relationship between the dependent variable y andone or more independent variables (or explanatory variables) x. Whenlinear regression is based on one explanatory variable, it is calledsimple linear regression, and when linear regression is based on two ormore explanatory variables, it is called multiple linear regression.

Linear regression models a regression equation using a linear predictorfunction, and unknown parameters are estimated from data. The regressionequation created in this way is called a linear model. There are severaluse cases of the linear regression, but linear regression may begenerally summarized into one of two categories.

First, if the goal is to predict a value, a predictive model suitablefor data is developed using the linear regression. By using thedeveloped linear regression equation, it is possible to predict y for avalue of x for which y is not given. In addition, when the dependentvariable y and associated independent variables X₁, . . . , X_(p)thereof exist, linear regression analysis may be used to quantify therelationship between X_(j) and y. X_(j) may have nothing to do with y,or may be a variable that provides additional information.

RELATED LITERATURE Patent Literature

-   Korean Patent Application Publication No. 10-2000-0058902 (Oct. 5,    2000)

SUMMARY

The present disclosure provides a parameter estimating method based oncorrection of independence of an error term, a computer program forimplementing the method, and a system configured to perform the method,the method which is capable of statistically estimating a measurementunit of a speedometer and accuracy thereof, the speedometer having anerror and measures a velocity of an object in an unknown measurementunit under the special theory of relativity.

The present disclosure also provides a parameter estimating method basedon correction of independence of an error term, a computer program forimplementing the method, and a system configured to perform the method,the method being capable of solving a problem that when the measurementunit of the speedometer is estimated by the least square method based ona linear regression model, the estimated value of the measurement unitand a variance of the estimated value vary depending on a velocity of ameasuring person (whereby the confidence interval and the statisticalconfidence of the estimated value of the measurement unit vary).

The present disclosure also provides a parameter estimating method basedon correction of independence of an error term, a computer program forimplementing the method, and a system configured to perform the method,the method which estimates a parameter using an independencerelationship between a momentum of an object measured at therelativistic center of momentum and the error term, so that theestimated value of the measurement unit and the variance of theestimated value (accordingly, the confidence interval and statisticalsignificance of the estimated value of the measurement unit) can beacquired regardless of a velocity of a measuring person.

The present disclosure also provides a parameter estimating method basedon correction of independence of an error term, a computer program forimplementing the method, and a system configured to perform the method,the method being capable of being effectively applied even to other datahaving the relationship between a measured velocity with an error of anunknown measurement unit under the special theory of relativity and anactual velocity.

In an aspect, there is provided is a parameter estimating method basedon correction of independence of an error term, and the method includes:acquiring a measured velocity, which is a velocity of a moving objectmeasured with a speedometer being operated by an observer and having aspecific measurement unit, and an actual velocity of the moving objectwith respect to the observer; performing modeling of a linear regressionmodel with the measured velocity as a dependent variable and the actualvelocity of the moving object with respect to the observer as anindependent variable; and estimating parameters of a linear regressionequation after correcting the independence of the error term based onLorentz velocity transformation.

The actual velocity may include a velocity of a moving object measuredwith a speedometer having a negligible error, and the acquiring of thevelocity may include measuring the velocity of the moving object with aspeedometer having an unknown measurement unit as the specificmeasurement unit and having an error in velocity measurement, the errorexcluding the negligible error.

The estimating of the parameters may include specifying the independenceof the error term with respect to a population regression equationdefined by Equation 1 below as the linear regression equation:

Y _(i)=β₀+β₁ ·X _(i)+∈_(i)  [Equation 1]

where Y_(i) denotes a velocity of moving object i measured with thespeedometer, X_(i) denotes an actual velocity of the moving object i (ora velocity of the moving object i measured with the speedometer havingthe negligible error), β₀ denotes accuracy or a systematic error of thespeedometer, β₁ denotes the measurement unit of the speedometer, and εidenotes an error term that follows N(0,σ²) (where N(0,σ²) is a normaldistribution with a variance of σ²).

The estimating of the parameters may include setting the independentvariable to be bounded to an open interval (−c, c) (where c is anyfinite value).

The estimating of the parameters may include correcting the independenceof the error term to E[ε·f(φ)]=0 by setting every moving object to havesame rest mass in the Lorentz velocity transformation and then applyingrapidity φ of a moving object measured at a relativistic center ofmomentum.

The estimating of the parameters may include deriving E[ε]=0 andE[ε·sinh(φ)]=0 as a population moment condition based on the correctedindependence of the error term, and estimating the parameters of thelinear regression equation using the population regression equation andthe derived population moment condition.

The estimating of the parameters may include estimating the parametersof the linear regression equation after deriving a sample regressionequation and a sample moment condition, which correspond to thepopulation regression equation and the derived population momentcondition.

The estimating of the parameters may include estimating accuracy and themeasurement unit of the speedometer as the parameters of the linearregression equation through Equation 2 below:

$\begin{matrix}{{{\hat{\beta}}_{0} = {{\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {\overset{\_}{Y} - {\frac{\sum_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} \cdot \overset{\_}{X}}}}}{{\hat{\beta}}_{1} = \frac{\sum_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

where {circumflex over (β)}₀ denotes an estimated accuracy of thespeedometer, {circumflex over (β)}₁ denotes an estimated measurementunit of the speedometer,

${\overset{¯}{X} = {\frac{1}{N}{\sum_{i = 1}^{N}X_{i}}}},{\overset{¯}{Y} = {\frac{1}{N}{\sum_{i = 1}^{N}Y_{i}}}},{\phi_{i} = {\theta_{i} - \theta_{0}}},{\theta_{i} = {\tanh^{- 1}\left( \frac{X_{i}}{C} \right)}},{and}$${\tanh\left( \theta_{0} \right)} = {\frac{\sum_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}.}$

The estimating of the parameters may include, based on the estimatedparameters of the linear regression equation, estimating the respectivevariances for the corresponding parameters through Equation 3 below:

$\begin{matrix}{{\begin{matrix}{{{Var}\left( {\hat{\beta}}_{0} \right)} = {{{Var}\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} \right)} = {{{Var}\left( \overset{\_}{Y} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}}} \\{= {{{Var}\left( {\frac{1}{N}{\sum\limits_{i = 1}^{N}\;\left( {\beta_{0} + {\beta_{1} \cdot X_{i}} + \epsilon_{i}} \right)}} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{\frac{1}{N^{2}} \cdot N \cdot \sigma^{2}} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {\frac{1}{N}\left( {1 + \frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right){\overset{\_}{X}}^{2}}{2\;{c^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} \right)\sigma^{2}}}\end{matrix}{{Var}\left( {\hat{\beta}}_{1} \right)}} = {{\sigma^{2}\frac{N\left( {T - 1} \right)}{2}\frac{S^{2} - C^{2}}{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} = {\frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right)}{2{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}}\sigma^{2}}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

where Var({circumflex over (β)}₀) denotes a variance for {circumflexover (β)}₀, Var({circumflex over (β)}₁) denotes a variance for{circumflex over (β)}₁,

${S = {\frac{1}{N}{\sum_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}}},{C = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}}},{T = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\left( {2 \cdot \phi_{i}} \right)}}}},{and}$$H = {\frac{N}{\sum_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}}.}$

The estimating of the parameters may include, based on the estimatedparameters of the linear regression equation, estimating the covariancebetween the corresponding parameters through Equation 4 below:

$\begin{matrix}\begin{matrix}{{{Cov}\left( {{\hat{\beta}}_{0},{\hat{\beta}}_{1}} \right)} = {E\left\lbrack {\left( {{\hat{\beta}}_{0} - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right)\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - \beta_{1}} \right)} \right\rbrack}} \\{= {E\left\lbrack {{- \overset{\_}{X}} \cdot \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2}} \right\rbrack}} \\{= {{- \overset{\_}{X}} \cdot {E\left\lbrack \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2} \right\rbrack}}} \\{= {{- \overset{\_}{X}} \cdot {{Var}\left( {\hat{\beta}}_{1} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

where Cov({circumflex over (β)}₀,{circumflex over (β)}₁) denotes acovariance between {circumflex over (β)}₀ and {circumflex over (β)}₁.

In another aspect, there is provided a parameter estimating system basedon correction of independence of an error term, and the system includes:a moving object velocity acquiring unit configured to acquire a measuredvelocity, which is a velocity of a moving object measured with aspeedometer being operated by an observer and having a specificmeasurement unit, and an actual velocity of the moving object withrespect to the observer; a modeling unit configured to perform modelingof a linear regression model with the measured velocity as a dependentvariable and the actual velocity of the moving object with respect tothe observer as an independent variable; and a parameter estimating unitconfigured to estimate parameters of a linear regression equation aftercorrecting the independence of the error term based on Lorentz velocitytransformation.

In yet another aspect, there is provided a computer program stored in acomputer-readable recording, the program in which each step of theparameter estimating method based on correction of independence of anerror term is performed by a processor of an information processingdevice.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating a parameter estimating system accordingto an embodiment of the present disclosure.

FIG. 2 is a diagram illustrating a physical configuration of a parameterestimating apparatus in FIG. 1.

FIG. 3 is a diagram for explaining a functional configuration of aparameter estimating apparatus in FIG. 1.

FIG. 4 is a flowchart illustrating a process of estimating a parameterfor velocity measurement based on correction of independence of an errorterm according to an embodiment of the present disclosure.

DETAILED DESCRIPTION

The following descriptions about the present disclosure are merelyembodiments for describing the present disclosure in a structural orfunctional view and the scope of the invention should not be construedas being limited to the embodiments set forth herein. That is, sinceembodiments of the invention can be variously changed and have variousforms, the scope of the present disclosure should be understood toinclude equivalents capable of realizing the technical spirit. Further,it should be understood that since a specific embodiment should includeall objects or effects or include only the effects, the scope of thepresent disclosure is limited by the objects or effects.

Meanwhile, meanings of terms described in the present application shouldbe understood as follows.

The terms “first,” “second,”, and the like are used to differentiate acertain component from other components, but the scope of the presentdisclosure should not be construed to be limited by the terms. Forexample, a first component may be referred to as a second component, andsimilarly, the second component may be referred to as the firstcomponent.

It should be understood that, when it is described that a component is“connected to” another component, the component may be directlyconnected to another component or a third component may be presenttherebetween. In contrast, it should be understood that, when it isdescribed that an element is “directly connected to” another element, itis understood that no element is present between the element and anotherelement. Meanwhile, other expressions describing the relationship of thecomponents, that is, expressions such as “between” and “directlybetween” or “adjacent to” and “directly adjacent to” should be similarlyinterpreted.

It is to be understood that singular expressions encompass a pluralityof expressions unless the context clearly dictates otherwise and itshould be understood that term “include” or “have” indicates that afeature, a number, a step, an operation, a component, a part, or thecombination thereof described in the specification is present, but doesnot exclude a possibility of the presence or addition of one or moreother features, numbers, steps, operations, components, parts orcombinations thereof, in advance.

In each step, reference numerals (e.g., a, b, c, etc.) are used forconvenience of description, the reference numerals are not used todescribe the order of the steps and unless otherwise stated, it mayoccur differently from the order specified. That is, the respectivesteps may be performed similarly to the specified order, performedsubstantially simultaneously, and performed in an opposite order.

The present disclosure can be implemented as a computer-readable code ona computer-readable recording medium and the computer-readable recordingmedium includes all types of recording devices for storing data that canbe read by a computer system. Examples of the computer readablerecording medium include a ROM, a RAM, a CD-ROM, a magnetic tape, afloppy disk, an optical data storage device, and the like. Further, thecomputer readable recording media may be stored and executed as codeswhich may be distributed in the computer system connected through anetwork and read by a computer in a distribution method.

If it is not contrarily defined, all terms used herein have the samemeanings as those generally understood by those skilled in the art.Terms which are defined in a generally used dictionary should beinterpreted to have the same meaning as the meaning in the context ofthe related art, and are not interpreted as an ideal meaning orexcessively formal meanings unless clearly defined in the presentapplication.

FIG. 1 is a diagram illustrating a parameter estimating system accordingto an embodiment of the present disclosure.

Referring to FIG. 1, a parameter estimating system 100 may include auser terminal 110, a parameter estimating apparatus 130, and a database150.

The user terminal 110 may correspond to a computing device capable ofproviding analysis data for parameter estimation and checking theanalysis result, and may be implemented as a smartphone, a laptopcomputer, or a computer. However, aspects of the present disclosure arenot necessarily limited thereto, and the user terminal 110 may beimplemented as any of various devices, such as a tablet PC and the like.The user terminal 110 may be connected to the parameter estimatingapparatus 130 via a network, and a plurality of user terminals 110 maybe connected to the parameter estimating apparatus 130 at the same time.

The parameter estimating apparatus 130 may be implemented as a servercorresponding to a computer or program that performs an operation ofestimating a measurement unit and accuracy of a speedometer under thespecial theory of relativity and provides the result of the estimation.The parameter estimating apparatus 130 may be connected to the userterminal 110 via a network and may exchange information therewith.

In one embodiment, the parameter estimating apparatus 130 may storenecessary data in a process of estimating a parameter of a linearregression model by compensating independence of an error term of thelinear regression model based on analysis data that is collected inconnection with the database 150. Meanwhile, unlike FIG. 1, theparameter estimating apparatus 130 may include the database 150. Inaddition, the parameter estimating apparatus 130 may be a physicalcomponent forming the system and may include a processor, a memory, auser input/output unit, and a network input/output unit, which will bedescribed in more detail in FIG. 2.

The database 150 may correspond to a storage device that stores varioustypes of information required to operate the parameter estimatingapparatus 130. The database 150 may store analysis data for parameterestimation, and may store information on a linear regression model basedon the analysis data. However, aspects of the present disclosure are notnecessarily limited thereto, and the database 150 may store informationthat is collected or processed in various forms during a process inwhich the parameter estimating apparatus 130 estimates a parameter basedon correction of independence of an error term.

FIG. 2 is a diagram illustrating a physical configuration of a parameterestimating apparatus in FIG. 1.

Referring to FIG. 2, the parameter estimating apparatus 130 may includea processor 210, a memory 230, a user input/output unit 250, and anetwork input/output unit 270.

The processor 210 may execute a procedure for processing each step inthe process of operating the parameter estimating apparatus 130, maymanage the memory 230 to be read or written throughout the process, andmay schedule the synchronization time between a volatile memory and anonvolatile memory in the memory 230. The processor 210 may control theoverall operation of the parameter estimating apparatus 130, and may beelectrically connected to the memory 230, the user input/output unit250, and the network input/output unit 270 to control data flowtherebetween. The processor 210 may be implemented as a centralprocessing unit (CPU) of the parameter estimating apparatus 130.

The memory 230 may include an auxiliary memory device implemented as anonvolatile memory such as a solid state drive (SSD) or a hard diskdrive (HDD) to store all data required for the parameter estimatingapparatus 130, and the memory 230 may include a main memory deviceimplemented as a volatile memory such as random access memory (RAM).

The user input/output unit 250 may include an environment for receivinga user input and an environment for outputting specific information to auser. For example, the user input/output unit 250 may include an inputdevice with an adapter, such as a touch pad, a touch screen, anon-screen keyboard, or a pointing device, and may also include an outputdevice with an adapter, such as a monitor or a touch screen. In oneembodiment, the user input/output unit 250 may correspond to a computingdevice that is connected through remote connection, and in this case,the parameter estimating apparatus 130 may serve as a server.

The network input/output unit 270 may include an environment forconnecting to an external device or system via a network and mayinclude, for example, an adapter for communication, such as a local areanetwork (LAN), a metropolitan area network (MAN), a wide area network(WAN), a VAN (Value Added Network), and the like.

FIG. 3 is a diagram for explaining a functional configuration of aparameter estimating apparatus in FIG. 1.

Referring to FIG. 3, the parameter estimating apparatus 130 may includea moving object velocity acquiring unit 310, a modeling unit 330, aparameter estimating unit 350, and a controller 370.

The moving object velocity acquiring unit 310 may acquire a measuredvelocity, which is a velocity of a moving object measured with aspeedometer being operated by an observer and having a specificmeasurement unit, and an actual velocity of the moving object withrespect to the observer. In one embodiment, the moving object velocityacquiring unit 310 may include a measurement unit in an unidentifiedstate as the specific measurement unit, and may acquire a measuredvelocity by acquiring a velocity of a moving object with a speedometerhaving an error in velocity measurement, the error excluding anegligible error. In this case, the actual velocity of the moving objectwith respect to the observer may include a velocity of the movingobject, which is measured with the speedometer having the negligibleerror.

Meanwhile, the moving object velocity acquiring unit 310 may receive ameasured velocity and an actual velocity of a specific moving objectthrough the user terminal 110 or retrieve data stored in the database150, thereby being enabled to acquire the corresponding information.

That is, the parameter estimating apparatus 130 may statisticallyestimate a velocity measurement unit of a speedometer and accuracythereof, the speedometer having an error and measuring a velocity of anobject in an unknown measurement unit.

The modeling unit 330 may perform modeling of a linear regression modelwith the measured velocity as a dependent variable and the actualvelocity of the moving object with respect to the observer as anindependent variable.

The parameter estimating unit 350 may estimate parameters of a linearregression equation after correcting independence of an error term basedon Lorentz velocity transformation. In one embodiment, the parameterestimating unit 350 may set the independent variable to be bounded to anopen interval (−c, c) (where c is any finite value). That is, theparameter estimating unit 350 may correct the independence of the errorterm, considering the special theory of relativity in which if c denotesthe speed of light, a velocity of an object can be observed only withinthe interval of (−c,c) in the real world and a velocity of the sameobject with respect to other observers is transformed by the Lorentzvelocity transformation.

Meanwhile, although the present disclosure has been described with theexample in which c denotes the speed of light, aspects of the presentdisclosure are not necessarily limited thereto and c may be any finitevalue according to a measurement unit of a speedometer that measures anactual velocity of an object.

More specifically, the parameter estimating unit 350 may correct anerror term in a way that satisfies a condition in which the error termis independent not just of the exact velocity of the object, but also ofthe velocity of a measuring person, and the correction may beindependent of the velocity of the measuring person and may be processedbased on a result of establishing an independence relationship betweenthe error term and a value specifying the velocity of the object. Tothis end, in a case where the rest mass of each moving object i isconstant, the parameter estimating unit 350 may estimate a parameter byutilizing the independence relationship between the error term andmomentum of an object measured at the relativistic center of momentum.

In one embodiment, the parameter estimating unit 350 may specify theindependence of the error term with respect to a population regressionequation that is defined by Equation 1 below as a linear regressionequation. That is, the parameter estimating apparatus 130 may constructa linear regression model, thereby being enabled to statisticallyestimate the velocity measurement unit of the speedometer and theaccuracy thereof as parameters by the least square method.

Y _(i)=β₀+β₁ ·X _(i)+∈_(i)  [Equation 1]

Here, Y_(i) denotes a velocity of moving object i measured with aspeedometer, X_(i) denotes an actual velocity of the moving object i (ora velocity of the moving object i measured with the speedometer having anegligible error), β₀ denotes accuracy or a systematic error of thespeedometer, β₁ denotes a measurement unit of the speedometer, and εidenotes an error term that follows N(0,σ²).

If X₁ corresponds to the velocity of the moving object i measured withthe speedometer having the negligible error, Y_(i) may correspond to thevelocity of the moving object i measured using a speedometer having anerror in velocity measurement, the error excluding the negligible error.

In one embodiment, the parameter estimating unit 350 may correct theindependence of the error term to E[ε·f(φ)]=0 by setting every movingobject to have the same rest mass in the Lorentz velocity transformationand then applying rapidity φ of a moving object measured at therelativistic center of momentum.

In one embodiment, the parameter estimating unit 350 may derive E[ε]=0and E[ε·sinh(φ)]=0 as a population moment condition based on thecorrected independence of the error term, and may estimate theparameters of the linear regression equation using the populationregression equation and the derived population moment condition.

In one embodiment, the parameter estimating unit 350 may estimate theparameters of the linear regression equation after deriving a sampleregression equation and a sample moment condition, which correspond tothe population regression equation and the derived population momentcondition.

In one embodiment, the parameter estimating unit 350 may estimate theaccuracy and the measurement unit of the speedometer as the parametersof the linear regression equation through Equation 2 below.

$\begin{matrix}{{{\hat{\beta}}_{0} = {{\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {\overset{\_}{Y} - {\frac{\sum_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} \cdot \overset{\_}{X}}}}}{{\hat{\beta}}_{1} = \frac{\sum_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$

Here, {circumflex over (β)}₀ denotes the estimated accuracy of thespeedometer, and {circumflex over (β)}₁ denotes the estimatedmeasurement unit of the speedometer. Also,

${\overset{¯}{X} = {\frac{1}{N}{\sum_{i = 1}^{N}X_{i}}}},{\overset{¯}{Y} = {\frac{1}{N}{\sum_{i = 1}^{N}Y_{i}}}},{\phi_{i} = {\theta_{i} - \theta_{0}}},{\theta_{i} = {\tanh^{- 1}\left( \frac{X_{i}}{c} \right)}},{and}$${\tanh\left( \theta_{0} \right)} = {\frac{\sum_{i - 1}^{N}{\sinh\left( \theta_{i} \right)}}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}.}$

In one embodiment, based on the estimated parameters of the linearregression equation, the parameter estimating unit 350 may estimatevariances and a covariance for the corresponding parameters.

In one embodiment, based on the estimated parameters of the linearregression equation, the parameter estimating unit 350 may estimate avariance for the corresponding parameters through Equation 3 below.

$\begin{matrix}{\begin{matrix}{{{Var}\left( {\hat{\beta}}_{0} \right)} = {{{Var}\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} \right)} = {{{Var}\left( \overset{\_}{Y} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}}} \\{= {{{Var}\left( {\frac{1}{N}{\sum\limits_{i = 1}^{N}\;\left( {\beta_{0} + {\beta_{1} \cdot X_{i}} + \epsilon_{i}} \right)}} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{\frac{1}{N^{2}} \cdot N \cdot \sigma^{2}} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {\frac{1}{N}\left( {1 + \frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right){\overset{\_}{X}}^{2}}{2\;{c^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} \right)\sigma^{2}}}\end{matrix}{{{Var}\left( {\hat{\beta}}_{1} \right)} = {{\sigma^{2}\frac{N\left( {T - 1} \right)}{2}\frac{S^{2} - C^{2}}{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} = {\frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right)}{2{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}}\sigma^{2}}}}} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack\end{matrix}$

Here, Var({circumflex over (β)}₀) denotes the variance for {circumflexover (β)}₀, and Var({circumflex over (β)}₁) denotes the variance for{circumflex over (β)}₁. Also,

${S = {\frac{1}{N}{\sum_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}}},{C = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}}},{T = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\;\left( {2 \cdot \phi_{i}} \right)}}}},{and}$$H = {\frac{N}{\sum_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}}.}$

In one embodiment, based on the estimated parameters of the linearregression equation, the parameter estimating unit 350 may estimate thecovariance between the corresponding parameters through Equation 4below.

$\begin{matrix}\begin{matrix}{{{Cov}\left( {{\hat{\beta}}_{0},{\hat{\beta}}_{1}} \right)} = {E\left\lbrack {\left( {{\hat{\beta}}_{0} - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right)\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - \beta_{1}} \right)} \right\rbrack}} \\{= {E\left\lbrack {{- \overset{\_}{X}} \cdot \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2}} \right\rbrack}} \\{= {{- \overset{\_}{X}} \cdot {E\left\lbrack \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2} \right\rbrack}}} \\{= {{- \overset{\_}{X}} \cdot {{Var}\left( {\hat{\beta}}_{1} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$

Here, Cov({circumflex over (β)}₀,{circumflex over (β)}₁) denotes thecovariance between {circumflex over (β)}₀ and {circumflex over (β)}₁.

The controller 370 may control the overall operation of the parameterestimating apparatus 130, and may manage a control flow or data flowbetween the moving object velocity acquiring unit 310, the modeling unit330, and the parameter estimating unit 350.

FIG. 4 is a flowchart illustrating a process of estimating a parameterfor velocity measurement based on correction of independence of an errorterm according to an embodiment of the present disclosure.

Referring to FIG. 4, the parameter estimating apparatus 130 may use themoving object velocity acquiring unit 310 to acquire a measuredvelocity, which is a velocity of a moving object measured with aspeedometer being operated by an observer and having a specificmeasurement unit, and an actual velocity of the moving object withrespect to the observer in operation S410. The parameter estimatingapparatus 130 may use the modeling unit 330 to perform modeling of alinear regression model with the measured velocity as a dependentvariable and the actual velocity of the moving object with respect tothe observer as an independent variable in operation S430. In a casewhere the independent variable is bounded to an open interval, that is,if X∈(−c,c) (where c is any finite value), the parameter estimatingapparatus 130 may use the parameter estimating unit 350 to estimate theparameters of the linear regression equation after correctingindependence of an error term for the linear regression equation basedon the Lorentz velocity transformation, in operation S453.

On the contrary, in a case where the independent variable is not boundedto the open interval, that is, if X∈(−∞,∞), the parameter estimatingapparatus 130 may use the parameter estimating unit 350 to estimate theparameters of the linear regression equation after specifying theindependence of the error term for the linear regression equation basedon Galilean velocity transformation, in operation S451. In this case, anOLS estimator and a variance thereof may be utilized.

Hereinafter, a parameter estimating method based on correction ofindependence of an error term according to the present disclosure willbe described in more detail.

1. INTRODUCTION

Linear regression is one of the most frequently used models in empiricalanalysis.

Y _(i)=β₀+β₁ ·X _(1i)+β₂ ·X _(2i)+ . . . +β_(p) ·X _(pi)+∈_(i)

Here, β_(p) denotes a measure of association between the independentvariable X_(p) and the dependent variable Y, and β₀ denotes an expectedvalue of Y when all X_(p) are equal to 0. ε_(i) denotes an error term.

The linear regression model is based on the following assumptions.

Assumption 1: The independent variable is measured without an error.

Assumption 2: The error is independent of the independent variable.

Assumption 3: The error is independently and identically normallydistributed.

In a linear regression model, an unknown parameter is often estimatedusing the ordinary least squares (OLS) method. This is because anOrdinary Least Squares (OLS) estimator has desirable properties as aparameter estimator such as unbiasedness, consistency, and efficiency.

The linear regression model and the OLS estimator provide accurateinference and an accurate estimated value only when the aboveassumptions are true. The assumption of an error term that is normallydistributed conditional on the independent variables implies that thedependent variable can be any real number. When the range of dependentvariables is bounded (for example, when the variables are discontinuousor bounded), this assumption is violated. Since a linear regressionmodel with a bounded dependent variable can lead to a serious error ininference, alternative nonlinear models and procedures have beendeveloped and adopted, for example, a Tobit model regarding a censoreddependent variable and a Poisson regression model regarding a count(non-negative integer) dependent variable.

In contrast, researchers do not pay attention to whether a bounded rangeof independent variables exists in a model as long as the variables areexogenous and measured without an error. This is because such variablesdo not violate any assumptions of the linear regression model.Traditional (e.g., OLS) estimators are used to estimate unknownparameters of the linear regression model although the range ofindependent variables is limited.

Does any problem arise with the use of the OLS estimators when the rangeof independent variables is bounded as long as the independent variablesare exogenous and measured without an error? If so, what is the desiredestimator when the range of independent variables is bounded?

A simple linear regression model in which exogenous and error-freeindependent variables are essentially bounded to an open interval isexplored in order to investigate a problem that occurs when an OLSestimator is used in a case where the range of independent variables isbounded. A linear regression model is used to estimate a unit (scale)(i.e., a slope coefficient) and accuracy (i.e., an interceptcoefficient) of the speedometer according to special relativity (seeChapter 2 for details). In this model, the dependent variable is avelocity of an object measured by an observer with a speedometer havinga normally distributed error, and the independent variable is an actualvelocity of the object with respect to the observer. In the real worldwhere the special theory of relativity is applied, the actual velocity(independent variable) of the object (which has mass, i.e., static massgreater than 0) with respect to the observer is bounded to the openinterval, (−c,c). Here, c denotes the speed of light.

The OLS estimator for the slope coefficient depends on the observer'svelocity under special relativity. In order to address this problem, thepresent disclosure proposes and uses a new estimator for a slopecoefficient that is independent of the observer's velocity. In addition,an estimator for a new intercept coefficient is also proposed and used.

It is confirmed that the proposed estimator is an unbiased estimator,and the proposed estimator converges to the OLS estimator when capproaches infinity. The variance of the proposed estimator is greaterthan that of the OLS estimator, which reflects the fact that when therange of independent variables is bounded, the uncertainty of theestimated slope coefficient and the estimated intercept coefficientbecomes greater. When c approaches infinity, the variance of theproposed estimator converges to the variance of the OLS estimator.

Hereinafter, the remaining parts are configured as follows. Chapter 2explains a linear regression model according to special relativity.Chapter 3 explains that it is inadequate to use an OLS estimator in alinear regression model under the special theory of relativity. Chapter4 provides the rationale for an alternative estimator. Chapter 5explains an alternative estimator for a linear regression model underthe special theory of relativity. Chapter 6 explains properties of analternative estimator. Lastly, Chapter 7 provides a summary of theconclusion.

2. LINEAR REGRESSION MODEL UNDER SPECIAL RELATIVITY

This chapter explains a special relativity situation in which a simplelinear regression model with an independent variable bounded to anessentially open interval (−c,c) appears.

Suppose that an engineer has developed a speedometer. The speedometerhas 0 on average in terms of precision, and the variance of thespeedometer has a random error that follows an unknown normaldistribution. The random error is independent of the actual velocity ofa measured object, as well as the velocity of an observer who iscarrying the speedometer. The engineer cannot be sure in what unit thespeedometer measures the velocity of the object, such as meters/seconds,miles/hours, or the like. If the actual velocity of the object ismeasured in meters/second, the measurement unit of the newly developedspeedometer may be expressed as β₁ meters/second. In the worst case, thespeedometer does not reflect the actual velocity of the object at all,i.e., β₁=0. Also, when measuring the velocity of a stationary objectfrom the observer, the engineer is not sure if the speedometer isadjusted to 0, that is, if the speedometer shows the velocity of zero.That is, the speedometer has a systematic bias β₀ in terms of accuracy.

Suppose that the engineer brings a speedometer to a researcher who has aspeedometer (or a speedometer having a negligible error) that canaccurately measure the velocity of an object in meters/second. Theengineer asks the researcher about whether his speedometer can measurethe velocity of the object (that is, whether β₁≠0), and, if so (that is,if β₁≠0), what scale the speedometer uses (what measurement unit is),and how much the engineer has to adjust his speedometer to ensure thevelocity of 0 with respect to a stationary object from an observer. Inthe real (relativistic) world, the actual velocity of the object (whichhas mass, i.e., static mass greater than 0) is bounded to the openinterval (−c,c). Here, c denotes the speed of light in meters/second.Therefore, the researcher's speedometer always shows values in the rangeof (−c,c).

Therefore, a population regression equation may be expressed as follows.

Y _(i)=β₀+β₁ ·X _(i)+∈_(i)  (2.1)

Y_(i): velocity of object i measured with a newly developed speedometer

X_(i): actual velocity of the object i measured in meters/second,X_(i)∈(−c,c)

β₀: systematic error of the speedometer with respect to a stationaryobject from an observer

β₁: scale of the speedometer in meters/second (measurement unit)

εi: error term of the speedometer, which follows N(0,σ²) (normaldistribution)

3. INVESTIGATING THE INADEQUACY OF OLS ESTIMATOR UNDER SPECIALRELATIVITY

This chapter explains that it is inappropriate to use an OLS estimatorunder special relativity.

3.1. Newtonian Universe Suppose that an engineer and a researcher livein the Newtonian universe. The actual velocity of an object (which hasmass, i.e., static mass greater than 0) may range from −∞ to ∞ in theNewtonian universe, so the regression model is a simple linearregression model with an unbounded range of independent variables.

The researcher may carry out experiments to measure the velocity of Nobjects moving along a straight line with a newly developed speedometerand an accurate speedometer (or a speedometer having a negligibleerror). The researcher may use an OLS estimator to estimate unknownparameters based on relevant data.

An OLS sample regression equation corresponding to Equation (2.1) may bewritten as follows.

Y _(i)={circumflex over (β)}_(0,OLS)+{circumflex over (β)}_(1,OLS) ·X_(i)+{circumflex over (∈)}_(i)  (3.1)

Here, {circumflex over (β)}_(0,OLS) and {circumflex over (β)}_(1,OLS)are the OLS estimators of β₀ and β₁, respectively, and {circumflex over(∈)}_(i) is the OLS residuals of the sample i.

The primary conditions of the OLS estimators {circumflex over(β)}_(0,OLS) and {circumflex over (β)}_(1,OLS) are Σ_(i=1) ^(N){circumflex over (∈)}_(i)=0 and Σ_(i=1) ^(N) {circumflex over(∈)}_(i)·X_(i)0, respectively.

The OLS estimator may be considered a method of moments estimator basedon the population moment conditions E[ε]=0 and E[ε·X]=0.

The estimated values {circumflex over (β)}_(1,OLS) and {circumflex over(β)}_(1,OLS) are bivariate normally distributed, and the means,variances, and covariance thereof are as follows.

$\begin{matrix}{{{\hat{\beta}}_{1,{OLS}} = {\frac{\sum\limits_{i = 1}^{N}{\left( {Y_{i} - \overset{\_}{Y}} \right) \cdot \left( {X_{i} - \overset{\_}{X}} \right)}}{\sum\limits_{i = 1}^{N}\left( {X_{i} - \overset{\_}{X}} \right)^{2}}\mspace{205mu}(3.2)}}\mspace{14mu}} \\{{\hat{\beta}}_{0,{OLS}} = {\overset{\_}{Y} - {{{\hat{\beta}}_{1,{OLS}} \cdot \overset{\_}{X\;}}\mspace{290mu}(3.3)}}} \\{{{Var}\left( {\hat{\beta}}_{1,{OLS}} \right)} = {\frac{1}{\sum\limits_{i = 1}^{N}\left( {X_{i} - \overset{\_}{X}} \right)^{2}}\sigma^{2}\mspace{265mu}(3.4)}} \\{{{Var}\left( {\hat{\beta}}_{0,{OLS}} \right)} = {\frac{\sum\limits_{i = 1}^{N}X_{i}^{2}}{\sum\limits_{i = 1}^{N}{N\left( {X_{i} - \overset{\_}{X}} \right)}^{2}}\sigma^{2}\mspace{245mu}(3.5)}} \\{{{Cov}\left( {{\hat{\beta}}_{0,{OLS}},{\hat{\beta}}_{1,{OLS}}} \right)} = {\frac{\overset{\_}{X}}{\sum\limits_{i = 1}^{N}{\left( {X_{i} - \overset{\_}{X}} \right)^{2}\sigma^{2}}}{\mspace{259mu}(3.6)}}}\end{matrix}$

Here,

$\overset{\_}{X} = {{\frac{1}{N}{\sum\limits_{i = 1}^{N}{X_{i}\mspace{14mu}{and}\mspace{14mu}\overset{\_}{Y}}}} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{Y_{i}.}}}}$

{circumflex over (β)}_(1,OLS) denotes the scale of the speedometer, and{circumflex over (β)}_(0,OLS) denotes the estimated value of thesystematic error. The (simultaneous) confidence interval thereof may bedetermined by the variances and the covariance.

If the researcher measures the velocity of an object with an accuratespeedometer (or a speedometer having a negligible error) while moving ina relatively positive direction than before at a constant velocity v*,X_(i)′=X_(i)−v* may be obtained. This transformation is called theGalilean velocity transformation. The relationship between X_(i) andX_(i)′ may be expressed as follows.

X _(i)=(X _(i) −X )+ X=x _(i) +X   (3.7)

X _(i) ′=X _(i) −v*=x _(i) +X−v*=x _(i) +X′  (3.8)

Here, x_(i)=X_(i)−X and

${\overset{\_}{X}}^{\prime} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{X_{i}^{\prime}.}}}$

X_(i) and X_(i)′ have the same demeaned velocity x_(i). In other words,the demeaned velocity does not change in the Galilean velocitytransformation. In addition, the sum of the demeaned velocity is 0 and(Σ_(i=1) ^(N) x_(i)=0), X_(i), X_(i)′, and x_(i) have the same variance.

If the researcher measures the velocity of an object with a newlydeveloped speedometer while moving in a positive direction with aconstant velocity v* than before,Y_(i)′=Y_(i)−β₁·(X_(i)−X_(i)′)=Y_(i)−β₁·v* may be measured.

Since the error term must be independent of the researcher's velocityand the object's actual velocity, the independence of the error termfrom the actual velocity must be specified by the velocity x_(i)obtained by removing the unchanging mean from the researcher's velocity.Therefore, the independence of the error term from the object's actualvelocity and the researcher's actual velocity is specified as E[ε·x]=0.E[ε·x]=0 is equal to E[ε·X]=0, and the error term independent of x isalso independent of X and X′.

The OLS estimator {circumflex over (β)}′_(1,OLS) based on X_(i)′ andY_(i)′ and the variance Var({circumflex over (β)}′_(1,OLS)) thereof arethe same as {circumflex over (β)}_(1,OLS) and Var({circumflex over(β)}_(1,OLS)).

This shows that the OLS estimator for β₁ and the variance thereof do notchange regardless of the researcher's velocity in the Newtonianuniverse. In other words, the OLS estimator for β₁ and the variancethereof do not change in the Galilean velocity transformation.

3.2. Real Relativistic World

The real world is different from the Newtonian universe. In the real(relativistic) world, the actual velocity of an object (which has mass,i.e., static mass greater than 0) is bounded to the open interval(−c,c), where c is the speed of light. Also, if a researcher in the real(relativistic) world measures the velocity of the object while moving ina positive direction with a constant velocity v* than before, thefollowing velocity may be obtained according to the Lorentz velocitytransformation.

$\begin{matrix}{X_{i}^{''} = \frac{X_{i} - v_{*}}{1 - \frac{v_{*} \cdot X_{i}}{c^{2}}}} & (3.9) \\{Y_{i}^{''} = {y_{i} - {\beta_{1} \cdot \left( {X_{i} - X_{i}^{''}} \right)}}} & (3.10)\end{matrix}$

Unlike the Newtonian universe case, the OLS estimator {circumflex over(β)}_(1,OLS)″ based on X_(i) and Y_(i)″ and the variance Var({circumflexover (β)}″_(1,OLS)) thereof are different from {circumflex over(β)}_(1,OLS) and Var({circumflex over (β)}_(1,OLS)). This shows that theOLS estimator for pi and the variance thereof are not independent of theresearcher's velocity in the real (relativistic) world.

This problem occurs because the demeaned velocity is not constant in theLorentz velocity transformation. Since X_(i) and X_(i)″ have differentdemeaned velocities, i.e., x_(i)≠x_(i)″ (where x_(i)″=X_(i)″−X″ and

$\left. {{\overset{\_}{X}}^{''} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{X_{i}^{''}.}}}} \right),$

E[ε·X]=0 is not the same as E[ε·X″]=0.

This result shows that in order to obtain an estimated value of piindependent of the researcher's velocity and the variance of theestimated value in the real (relativistic) world, the independence ofthe error term from the object's velocity and the independence of theerror term from the researcher's velocity need to be simultaneouslyspecified as a quantity that does not change in the Lorentz velocitytransformation.

4. INDEPENDENCE OF ERROR TERM IN THE RELATIVISTIC UNIVERSE

In this chapter, there is proposed a method for finding an unchangingquantity (invariant) in the Lorentz velocity transformation andspecifying independence of an error term from an object's velocity andindependence of the error term from the researcher's velocity at thesame time. The goal is to correctly estimate parameters of a regressionmodel. Details about the concept of special relativity and the conceptof the Lorentz invariant are omitted.

In physics, rapidity θ of velocity X is defined as follows.

$\begin{matrix}{\theta = {\tanh^{- 1}\left( \frac{X}{c} \right)}} & (4.1)\end{matrix}$

Relativistic momentum and energy of an object with the velocity 8 andthe static mass m are defined as follows.

Momentum=m·sinh(θ)  (4.2)

Energy=m·cosh(θ)  (4,3)

Let's say that θ_(i), θ_(i)″, and θ* are rapidities of X₁, X₁″, and v*,respectively.

$\begin{matrix}{\theta_{i} = {{\tanh^{- 1}\left( \frac{X_{i}}{c} \right)}\mspace{509mu}(4.4)}} \\{\theta_{i}^{''} = {{\tanh^{- 1}\left( \frac{X_{i}^{''}}{c} \right)}\mspace{500mu}(4.5)}} \\{{\theta_{*} = {{\tanh^{- 1}\left( \frac{v_{*}}{c} \right)}\mspace{509mu}(4.6)}}\mspace{121mu}}\end{matrix}$

In this case, the following relationship is established between θ_(i),θ_(i)″, and θ*.

θ_(i)″=θ_(i)=θ*  (4.7)

Therefore, X_(i) and X_(i)″ may be expressed as follows.

X _(i) =c·tanh(θ_(i))  (4.8)

X _(i) ″=c·tanh(θ_(i)″)=c·tanh(θ_(i)−θ*)  (4.9)

Let's define θ₀ and θ₀″ as follows.

$\begin{matrix}\begin{matrix}{{\tanh\left( \theta_{0} \right)} = {\frac{\sum\limits_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}{\sum\limits_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}\mspace{425mu}(4.10)}} \\{{{\tanh\left( \theta_{0}^{''} \right)} = {\frac{\sum\limits_{i = 1}^{N}{\sinh\left( {\theta_{i}}^{''} \right)}}{\sum\limits_{i = 1}^{N}{\cosh\left( \theta_{i}^{''} \right)}}\mspace{410mu}(4.11)}}\mspace{664mu}(4.12)}\end{matrix} & \square\end{matrix}$

In this case, the following relationship is established between θ₀, θ₀″,and θ*.

θ*=θ₀−θ₀″  (4.13)

If ϕ_(i),=θ_(i)−θ₀, θ_(i) and θ_(i)″ may be expressed as follows.

θ_(i)=(θ_(i)−θ₀)+θ₀=ϕ_(i)+θ₀  (4.14)

θ_(i) =θ_(i)−θ*=θ_(i)−(θ₀−θ₀″)=(θ_(i)−θ₀)+θ₀″=ϕ_(i)+θ₀″  (4.15)

These results show that ϕ_(i) does not change regardless of the velocityof the researcher. Since ϕ_(i) does not change, all functions of ϕ_(i),especially the relativistic momentum sinh(ϕ_(i)) and the relativisticenergy cosh(ϕ_(i)), do not change regardless of the velocity of theresearcher. Also, the following relationship is established.

$\begin{matrix}{{\sum\limits_{i = 1}^{N}{\sinh\left( \phi_{i} \right)}} = {{\sum\limits_{i = 1}^{N}{\sinh\left( {\theta_{i} - \theta_{0}} \right)}} = {{\sum\limits_{i = 1}^{N}{\sinh\left( {\theta_{i}^{''} - \theta_{0}^{''}} \right)}} = 0}}} & (4.16) \\{{\sum\limits_{i = 1}^{N}{\cosh\left( \phi_{i} \right)}} = {{\sum\limits_{i = 1}^{N}{\cosh\left( {\theta_{i} - \theta_{0}} \right)}} = {{\sum\limits_{i = 1}^{N}{\cosh\left( {\theta_{i}^{''} - \theta_{0}^{''}} \right)}} = 0}}} & (4.17)\end{matrix}$

Equation (4.16) shows that the sum of the relativistic momentum of theobjects is 0, assuming that all the objects have the same static mass ifthe researcher measures the velocity (rapidity) of an object whilemoving in a positive direction with a constant rapidity θ₀ than before.In this assumption, the rapidity θ₀ is related to the relativisticcenter of momentum. Thus, ϕ_(i) may be regarded as the rapidity of theobject i measured at the relativistic center of momentum, assuming thatall the objects have the same static mass.

Since the error term must be independent of the researcher's velocity aswell as the object's actual velocity, the independence of the error termneeds to be specified as a quantity that does not change with respect tothe researcher's velocity. Therefore, the independence of the error termmay be specified as E[ε·f(ϕ)]=0.

Parameters may be estimated using the following sample moment conditioncorresponding to the population moment conditions E[ε]=0 andE[ε·f(ϕ)]=0.

$\begin{matrix}\begin{matrix}{{\overset{\_}{Y} - {\hat{\beta}}_{0}} = {{{\hat{\beta}}_{1} \cdot \overset{\_}{X}} = 0}} \\{{\frac{1}{N}\left\{ {{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {f\left( \phi_{i} \right)}}} - {{\hat{\beta}}_{1}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {f\left( \phi_{i} \right)}}}}} \right\}} = 0}\end{matrix} & (4.18)\end{matrix}$

In a case where β₁=0, if Σ_(i=1) ^(N) f(ϕ_(i))≠0, β₀ is not uniquelyidentified. Accordingly, Σ_(i=1) ^(N) f(ϕ_(i)) must be derived as 0.Therefore, E[ε·sinh(ϕ)]=0 is selected as the population momentcondition.

5. SPECIAL RELATIVISTIC LINEAR REGRESSION ESTIMATOR

A population regression equation is as shown in Equation (2.1).Population moment conditions are as follows.

E[∈]=0  (5.1)

E[∈·sinh(ϕ)]  (5.2)

A sample regression equation is as follows.

Y _(i)={circumflex over (β)}₀+{circumflex over (β)}₁ ·X _(i)+{circumflexover (∈)}_(i)  (5.3)

Meanwhile, sample moment conditions are as follows.

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{i = 1}^{N}{\hat{\epsilon}}_{i}}} = {0\mspace{455mu}(5.4)}} \\{{\frac{1}{N}{\sum\limits_{i = 1}^{N}{{\hat{\epsilon}}_{i} \cdot {\sinh\left( \phi_{i} \right)}}}} = {0\mspace{455mu}(5.5)}}\end{matrix}$

The following is derived as follows from Equation (5.4).

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{i = 1}^{N}{\hat{\epsilon}}_{i}}} = {{\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {Y_{i} - {\hat{\beta}}_{0} - {{\hat{\beta}}_{1} \cdot X_{i}}} \right)}} = {{{\frac{1}{N}{\sum\limits_{i = 1}^{N}Y_{i}}} - {\hat{\beta}}_{0} - {\frac{{\hat{\beta}}_{1}}{N}{\sum\limits_{i = 1}^{N}X_{i}}}} = {{\overset{\_}{Y} - {\hat{\beta}}_{0} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = 0}}}} & (5.6)\end{matrix}$

The following is derived as follows from Equation (5.5).

$\begin{matrix}{{\frac{1}{N}{\sum\limits_{i = 1}^{N}{{\hat{\epsilon}}_{i} \cdot {\sinh\left( \phi_{i} \right)}}}} = {{\frac{1}{N}{\sum\limits_{i = 1}^{N}{\left( {Y_{i} - {\hat{\beta}}_{0} - {{\hat{\beta}}_{1} \cdot X_{i}}} \right) \cdot {\sinh\left( \phi_{i} \right)}}}} = {{\frac{1}{N}\left\{ {{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}} - {{\hat{\beta}}_{0}{\sum\limits_{i = 1}^{N}{\sinh\left( \phi_{i} \right)}}} - {{\hat{\beta}}_{1}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}}} \right\}} = {{\frac{1}{N}\left\{ {{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}} - {{\hat{\beta}}_{1}{\sum\limits_{i = 1}^{N}{{X_{i} \cdot \sinh}\text{|}\left( \phi_{i} \right)}}}} \right\}} = 0}}}} & (5.7)\end{matrix}$

Let's consider Σ_(i=1) ^(N) X_(i)·|sinh(ϕ_(i))>0 (Refer to A.1 forproof).

Therefore, the following is derived.

$\begin{matrix}{{\hat{\beta}}_{1} = {\frac{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}\mspace{461mu}(5.8)}} \\{{\hat{\beta}}_{0} = {{\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {\overset{\_}{Y} = {{{- \frac{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}} \cdot \overset{\_}{X}}\mspace{211mu}(5.9)}}}}\end{matrix}$

6. PROPERTIES OF SPECIAL RELATIVISTIC LINEAR REGRESSION ESTIMATOR

In this chapter, the properties of the proposed estimator are described.

6.1 Linearity of {circumflex over (β)}₁

The estimator {circumflex over (β)}₁ may be expressed as a linearcombination of Y_(i)(i=1, . . . , N) which are the sample values of Y.Referring to Equation (5.8), {circumflex over (β)}₁=Σ_(i=1) ^(N)k_(i)·Y_(i) and

$k_{i} = {\frac{\sinh\left( \phi_{i} \right)}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}.}$

This means that {circumflex over (β)}₁ is normally distributed. Also,since {circumflex over (β)}₀=Y−{circumflex over (β)}₁·X is shown in(5.9), {circumflex over (β)}₀ is also normally distributed. Thus,{circumflex over (β)}₁ and {circumflex over (β)}₀ are bivariate normallydistributed.

6.2 Unbiasedness of {circumflex over (β)}₁ and {circumflex over (β)}₀

$\begin{matrix}{{{\sum\limits_{i = 1}^{N}k_{i}} = {\frac{\sum\limits_{i = 1}^{N}{\sinh\left( \phi_{i\;} \right)}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} = {0.\mspace{14mu}{and}}}}{{\sum\limits_{i = 1}^{N}{k_{i} \cdot X_{i}}} = {\frac{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i\;} \right)}}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} = 1.}}} & (6.1) \\\begin{matrix}{{\hat{\beta}}_{1} = {\sum\limits_{i = 1}^{N}{k_{i} \cdot Y_{i}}}} \\{= {\sum\limits_{i = 1}^{N}{k_{i}\left( {\beta_{0} + {\beta_{1} \cdot X_{i}} + \epsilon_{i}} \right)}}} \\{= {{\beta_{0}{\sum\limits_{i = 1}^{N}k_{i}}} + {\beta_{1}{\sum\limits_{i = 1}^{N}{k_{i} \cdot X_{i}}}} + {\sum\limits_{i = 1}^{N}{k_{i} \cdot \epsilon_{i}}}}} \\{= {\beta_{1} + {\sum\limits_{i = 1}^{N}{k_{i} \cdot \epsilon_{i}}}}}\end{matrix} & \; \\\begin{matrix}{{E\left\lbrack {\hat{\beta}}_{1} \right\rbrack} = {E\left\lbrack {\beta_{1} + {\sum\limits_{i = 1}^{N}{k_{i} \cdot \epsilon_{i}}}} \right\rbrack}} \\{= {{E\left\lbrack \beta_{1} \right\rbrack} + {E\left\lbrack {\sum\limits_{i = 1}^{N}{k_{i} \cdot \epsilon_{i}}} \right\rbrack}}} \\{= {\beta_{1} + {\sum\limits_{i = 1}^{N}{{k_{i} \cdot {E\left\lbrack \epsilon_{i} \middle| X_{i} \right\rbrack}}\mspace{14mu}{since}\mspace{14mu}\beta_{1}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{constant}\mspace{14mu}{and}}}}} \\{{the}\mspace{14mu} k_{i}\mspace{14mu}{are}\mspace{14mu}{random}} \\{= {{\beta_{1} + {\sum\limits_{i = 1}^{N}{{k_{i} \cdot 0}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack \epsilon_{i} \middle| X_{i} \right\rbrack}}}} = {0\mspace{14mu}{by}\mspace{14mu}{assumption}}}} \\{= \beta_{1}}\end{matrix} & (6.2)\end{matrix}$

Therefore, {circumflex over (β)}₁ is the unbiased estimator of β₁.

$\begin{matrix}{{\hat{\beta}}_{0} = {{\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {{\left( {\beta_{0} + {\beta_{1} \cdot \overset{\_}{X}} + \overset{\_}{\epsilon}} \right) - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {\beta_{0} + {\left( {\beta_{1} - {\hat{\beta}}_{1}} \right) \cdot \overset{\_}{X}} + \overset{\_}{\epsilon}}}}} & (6.3) \\\begin{matrix}{{E\left\lbrack {\hat{\beta}}_{0} \right\rbrack} = {E\left\lbrack {\beta_{0} + {\left( {\beta_{1} - {\hat{\beta}}_{1}} \right) \cdot \overset{\_}{X}} + \overset{\_}{\epsilon}} \right\rbrack}} \\{= {{E\left\lbrack \beta_{0} \right\rbrack} + {E\left\lbrack {\left( {\beta_{1} - {\hat{\beta}}_{1}} \right) \cdot \overset{\_}{X}} \right\rbrack} + {E\left\lbrack \overset{\_}{\epsilon} \right\rbrack}}} \\{= {\beta_{0} + {\overset{\_}{X} \cdot {E\left\lbrack \left( {\beta_{1} - {\hat{\beta}}_{1}} \right) \right\rbrack}} + {{E\left\lbrack \overset{\_}{\epsilon} \right\rbrack}\mspace{14mu}{since}\mspace{14mu}\beta_{0}\mspace{14mu}{is}\mspace{14mu} a\mspace{14mu}{constant}}}} \\{= {{\beta_{0} + {{\overset{\_}{X} \cdot {E\left\lbrack \left( {\beta_{1} - {\hat{\beta}}_{1}} \right) \right\rbrack}}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack \overset{\_}{\epsilon} \right\rbrack}}} = {0\mspace{14mu}{by}\mspace{14mu}{assumption}}}} \\{= {\beta_{0} + {\overset{\_}{X}\left( {E\left\lbrack {\left( \beta_{1} \right\rbrack - {E\left\lbrack {\hat{\beta}}_{1} \right)}} \right\rbrack} \right)}}} \\{= {{\beta_{0} + {{\overset{\_}{X}\left( {\beta_{1} - \beta_{1}} \right)}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack \beta_{1} \right\rbrack}}} = {{\beta_{1}\mspace{14mu}{and}\mspace{14mu}{E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} = \beta_{1}}}} \\{= \beta_{0}}\end{matrix} & (6.4)\end{matrix}$

Therefore, {circumflex over (β)}₀ is the unbiased estimator of β₀.

6.3. Variance of {circumflex over (β)}₁ and {circumflex over (β)}₀

Using the assumption that yi is independently distributed, the varianceof {circumflex over (β)}₁ is as follows.

$\begin{matrix}\begin{matrix}{{{Var}\left( {\hat{\beta}}_{1} \right)} = {E\left\lbrack \left\{ {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right\}^{2} \right\rbrack}} \\{= {{{E\left\lbrack \left\{ {{\hat{\beta}}_{1} - \beta_{1}} \right\}^{2} \right\rbrack}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} = \beta_{1}}}\end{matrix} & (6.5)\end{matrix}$

Referring to Equation (6.1), it may be expressed as follows.

$\begin{matrix}{\left( {{\hat{\beta}}_{1} - \beta_{2}} \right)^{2} = {\left( {\sum\limits_{i = 1}^{N}{k_{i}\epsilon_{i}}} \right)^{2} = {{\sum\limits_{i = 1}^{N}{k_{i}^{2}\epsilon_{i}^{2}}} + {2{\sum\limits_{i = 1}^{N - 1}{\sum\limits_{j = {i + 1}}^{N}{k_{i}k_{j}\epsilon_{i}\epsilon_{j}}}}}}}} & (6.6)\end{matrix}$

Therefore, it may be expressed as follows.

$\begin{matrix}\begin{matrix}{{E\left\lbrack \left\{ {{\hat{\beta}}_{1} - \beta_{1}} \right\}^{2} \right\rbrack} = {E\left\lbrack {{\sum\limits_{i = 1}^{N}{k_{i}^{2}\epsilon_{i}^{2}}} + {2{\sum\limits_{i = 1}^{N - 1}{\sum\limits_{j = {i + 1}}^{N}{k_{i}k_{j}\epsilon_{i}\epsilon_{j}}}}}} \right\rbrack}} \\{= {{\sum\limits_{i = 1}^{N}{k_{i}^{2}{E\left\lbrack \epsilon_{i}^{2} \middle| X_{i} \right\rbrack}}} + {2{\sum\limits_{i = 1}^{N - 1}{\sum\limits_{j = {i + 1}}^{N}{k_{i}k_{j}{E\left\lbrack {\epsilon_{i}\epsilon_{j}} \middle| {X_{i}X_{j}} \right\rbrack}}}}}}} \\{{= {{\sum\limits_{i = 1}^{N}{k_{i}^{2}{E\left\lbrack \epsilon_{i}^{2} \middle| X_{i} \right\rbrack}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack {\epsilon_{i}\epsilon} \middle| {X_{i}X_{j}} \right\rbrack}}} = 0}}\mspace{14mu}} \\{{by}\mspace{14mu}{assumption}} \\{= {\sum\limits_{i = 1}^{N}{k_{i}^{2} \cdot \sigma^{2}}}} \\{= {{\sigma^{2}{\sum\limits_{i = 1}^{N}{k_{i}^{2}\mspace{14mu}{since}\mspace{14mu}{E\left\lbrack \epsilon_{i}^{2} \middle| X_{i} \right\rbrack}}}} = {\sigma^{2}\mspace{14mu}{by}}}} \\{assumption}\end{matrix} & (6.7) \\{{\sum\limits_{i = 1}^{N}k_{i}^{2}} = {\frac{1}{\left\{ {\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} \right\}^{2}}{\sum\limits_{i = 1}^{N}\left\{ {\sinh\left( \phi_{i} \right)} \right\}^{2}}}} & (6.8) \\\begin{matrix}{{\sum\limits_{i = 1}^{N}\left\{ {\sinh\left( \phi_{i} \right)} \right\}^{2}} = {\sum\limits_{i = 1}^{N}{\frac{1}{2}\left\{ {{\cosh\left( {2 \cdot \phi_{i}} \right)} - 1} \right\}}}} \\{= {{\frac{1}{2}{\sum\limits_{i = 1}^{N}{\cosh\left( {2 \cdot \phi_{i}} \right)}}} - \frac{N}{2}}} \\{= {\frac{N}{2}\left\{ {{\frac{1}{N}{\sum\limits_{i = 1}^{N}{\cosh\left( {2 \cdot \phi_{i}} \right)}}} - 1} \right\}}}\end{matrix} & (6.9) \\{{{\sum\limits_{i = 1}^{N}\left\{ {\sinh\left( \phi_{i} \right)} \right\}^{2}} = {\frac{N}{2}\left( {T - 1} \right)}}{{{where}\mspace{14mu} T} = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{\cosh\left( {2 \cdot \phi_{i}} \right)}}}}} & (6.10)\end{matrix}$

$\begin{matrix}\begin{matrix}{{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} = {\sum\limits_{i = 1}^{N}{c \cdot {\tanh\left( \theta_{i} \right)} \cdot {\sinh\left( {\theta_{i} - \theta_{0}} \right)}}}} \\{= {c{\sum\limits_{i = 1}^{N}{{\tanh\left( \theta_{i} \right)}\left\{ {{{\sinh\left( \theta_{i} \right)}{\cosh\left( \theta_{0} \right)}} -} \right.}}}} \\\left. {{\cosh\left( \theta_{i} \right)}{\sinh\left( \theta_{0} \right)}} \right\} \\{= {c{\sum\limits_{i = 1}^{N}\left\{ {{{\frac{\sinh\left( \theta_{i} \right)}{\cosh\left( \theta_{i} \right)} \cdot {\sinh\left( \theta_{i} \right)}}{\cosh\left( \theta_{0} \right)}} -} \right.}}} \\\left. {{\frac{\sinh\left( \theta_{i} \right)}{\cosh\left( \theta_{i} \right)} \cdot {\cosh\left( \theta_{i} \right)}}{\sinh\left( \theta_{0} \right)}} \right\} \\{= {c{\sum\limits_{i = 1}^{N}\left\{ {{{\cosh\left( \theta_{0} \right)}\frac{\sinh^{2}\left( \theta_{i} \right)}{\cosh\left( \theta_{i} \right)}} - {{\sinh\left( \theta_{0} \right)}{\sinh\left( \theta_{i} \right)}}} \right\}}}} \\{= {c{\sum\limits_{i = 1}^{N}\left\lbrack {{{\cosh\left( \theta_{0} \right)}\left\{ {{\cosh\left( \theta_{i} \right)} - \frac{1}{\cosh\left( \theta_{i} \right)}} \right\}} -} \right.}}} \\\left. {{\sinh\left( \theta_{0} \right)}{\sinh\left( \theta_{i} \right)}} \right\rbrack \\{= {c\left\lbrack {{{\cosh\left( \theta_{0} \right)}\left\{ {{\sum\limits_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}} - {\sum\limits_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}}} \right\}} -} \right.}} \\\left. {{\sinh\left( \theta_{0} \right)}{\sum\limits_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}} \right\rbrack\end{matrix} & (6.11)\end{matrix}$

Let's say,

$\begin{matrix}{{{C = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}}},{S = {\frac{1}{N}{\sum\limits_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}}},{and}}{H = {{\frac{N}{\sum\limits_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}}.{\tanh\left( \theta_{0} \right)}} = \frac{S}{C}}}} & (6.12) \\{{\cosh\left( \theta_{0} \right)} = {\frac{1}{\sqrt{1 - {\tan^{2}\left( \theta_{0} \right)}}} = {\frac{1}{\sqrt{1 - \frac{S^{2}}{C^{2}}}} = \frac{C}{\sqrt{S^{2} - C^{2}}}}}} & (6.13) \\{{\sinh\left( \theta_{0} \right)} = {{{\tanh\left( \theta_{0} \right)} \cdot {\cosh\left( \theta_{0} \right)}} = \frac{S}{\sqrt{S^{2} - C^{2}}}}} & (6.14) \\{{\sum\limits_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}} = \frac{N}{H}} & (6.15) \\\begin{matrix}{{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} = {c\left\lbrack {{\frac{C}{\sqrt{S^{2} - C^{2}}}\left\{ {{N \cdot C} - \frac{N}{H}} \right\}} -} \right.}} \\\left. {\frac{S2}{\sqrt{S^{2} - C^{2}}} \cdot N \cdot S} \right\rbrack \\{= {\frac{c \cdot N}{\sqrt{S^{2} - C^{2}}}\left\lbrack {C^{2} - S^{2} - \frac{C}{H}} \right\rbrack}}\end{matrix} & (6.16)\end{matrix}$

Therefore, it may be expressed as follows.

$\begin{matrix}{{{Var}\left( {\hat{\beta}}_{1} \right)} = {{\sigma^{2}\frac{N\left( {T - 1} \right)}{2}\frac{S^{2} - C^{2}}{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} = {\frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right)}{2{c^{2} \cdot {N\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}}\sigma^{2}}}} & (6.17)\end{matrix}$

The variance of {circumflex over (β)}₀ is as follows.

$\begin{matrix}\begin{matrix}{{{Var}\left( {\hat{\beta}}_{0} \right)} = {{Var}\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} \right)}} \\{= {{{Var}\left( \overset{\_}{Y} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{{Var}\left( {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {\beta_{0} + {\beta_{1} \cdot X_{i}} + \epsilon_{i}} \right)}} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{\frac{1}{N^{2}} \cdot N \cdot \sigma^{2}} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {\frac{1}{N}\left( {1 + \frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right){\overset{\_}{X}}^{2}}{2{c^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} \right)\sigma^{2}}}\end{matrix} & \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}(6.18) \\\;\end{matrix} \\\;\end{matrix} \\(6.19)\end{matrix} \\\;\end{matrix} \\\begin{matrix}(6.20) \\(6.21)\end{matrix}\end{matrix}\end{matrix}$

6.4. Covariance Between {circumflex over (β)}₁ and {circumflex over(β)}₀

The covariance between {circumflex over (β)}₁ and {circumflex over (β)}₀is as follows.

$\begin{matrix}\begin{matrix}{{{Cov}\left( {{\hat{\beta}}_{0},{\hat{\beta}}_{1}} \right)} = {E\left\lbrack {\left( {{\hat{\beta}}_{0} - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right)\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{{from}\mspace{14mu}{equation}\mspace{14mu}(5.9)} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{\left( {{{since}\mspace{14mu}{E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} = {{\overset{\_}{Y} - {{E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}\overset{\_}{X}}} = {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}}}} \right)} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - \beta_{1}} \right)} \right\rbrack}} \\{{from}\mspace{14mu}{equation}\mspace{14mu}(6.2)} \\{= {E\left\lbrack {{- \overset{\_}{X}} \cdot \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2}} \right\rbrack}} \\{= {{- \overset{\_}{X}} \cdot {E\left\lbrack \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2} \right\rbrack}}} \\{= {{- \overset{\_}{X}} \cdot {{Var}\left( {\hat{\beta}}_{1} \right)}}}\end{matrix} & (6.22)\end{matrix}$

7. SUMMARY AND CONCLUSIONS

The problem that occurs when OLS is used to estimate a linear regressionparameter in a case where independent variables are bounded to the openinterval (−c,c) in a special relativity environment has been described.It has been found that an OLS estimator for a slope parameter does notchange under the Lorentz velocity transformation.

As an alternative estimator for the parameter of the linear regressionunder special relativity, there has been proposed an estimator that doesnot change in the Lorentz velocity transformation. The proposedestimator is an unbiased estimator and converges to the OLS estimatorwhen c approaches infinity. When c approaches infinity, the variance ofthe proposed estimator also converges to the variance of the OLSestimator. Therefore, when the range of independent variables is boundedto the open interval, the proposed estimator may be regarded as ageneralization of the OLS estimator.

The variance of the proposed estimator is greater than that of the OLSestimator. This implies that the uncertainty is greater when the rangeof independent variables is bounded. Since the confidence intervalcomposed of the OLS estimator and the variance of the OLS estimator isnarrower than the confidence interval composed of the proposedestimator, if the range of independent variables is bounded in ahypothesis test that uses the OLS estimator and the variance of the OLSestimator, it is found that a liberal test could occur.

Meanwhile, the present disclosure has specifically described a parameterestimating method based on correction of independence of an error term,a computer program for implementing the method, and a system configuredto perform the method, the method which is capable of statisticallyestimating a measurement unit of a speedometer and accuracy thereof, thespeedometer which has an error and measures a velocity of an object inan unknown measurement unit under the special theory of relativity.However, aspects of the present disclosure are not necessarily limitedthereto, and of course, it is possible to effectively apply the presentdisclosure even to a different type of data that corresponds to therelationship between an actual velocity and a measured velocity with anerror of an unknown measurement unit under the special theory ofrelativity.

The disclosed technology may have the following effects. However, sincea specific embodiment may provide all the following effects or only afew of them, the scope of the disclosure is not limited to the followingadvantages.

A parameter estimating method based on correction of independence of anerror term according to an embodiment of the present disclosure, acomputer program for implementing the method, and a system configured toperform the method can solve the problem that when the measurement unitof the speedometer is estimated by the least square method based on alinear regression model, an estimated value of the measurement unit anda variance of the estimate (accordingly, the confidence interval andstatistical significance of the estimate) vary depending on a velocityof a measuring person.

A parameter estimating method based on correction of independence of anerror term according to an embodiment of the present disclosure, acomputer program for implementing the method, and a system configured toperform the method can make it possible to estimate a parameter using anindependence relationship between a momentum of an object measured atthe relativistic center of momentum and an error term, so that theestimated value of the measurement unit and the variance of the estimate(accordingly, the confidence interval and statistical significance ofthe estimate) can be acquired regardless of a velocity of a measuringperson.

A parameter estimating method based on correction of independence of anerror term according to an embodiment of the present disclosure, acomputer program for implementing the method, and a system configured toperform the method can be effectively applied even to other data havingthe relationship between a measured velocity with an error of an unknownmeasurement unit under the special theory of relativity and an actualvelocity.

While the present disclosure has been shown and described with referenceto preferred embodiments thereof, it will be understood by those skilledin the art that various changes in form and details may be made thereinwithout departing from the spirit and scope of the present disclosure asdefined by the appended claims.

What is claimed is:
 1. A parameter estimating method based on correctionof independence of an error term, the method comprising: acquiring ameasured velocity, which is a velocity of a moving object measured witha speedometer being operated by an observer and having a specificmeasurement unit, and an actual velocity of the moving object withrespect to the observer; performing modeling of a linear regressionmodel with the measured velocity as a dependent variable and the actualvelocity of the moving object with respect to the observer as anindependent variable; and estimating parameters of a linear regressionequation after correcting the independence of the error term based onLorentz velocity transformation.
 2. The method of claim 1, wherein theactual velocity comprises a velocity of a moving object measured with aspeedometer having a negligible error, and wherein the acquiring of thevelocity comprises measuring the velocity of the moving object with aspeedometer that has an unknown measurement unit as the specificmeasurement unit and has an error in velocity measurement, the errorwhich excludes the negligible error.
 3. The method of claim 1, whereinthe estimating of the parameters comprises specifying the independenceof the error term with respect to a population regression equationdefined by Equation 1 below as the linear regression equation:Y _(i)=β₀+β₁ ·X _(i)+∈_(i)  [Equation 1] where Yi denotes a velocity ofmoving object i measured with the speedometer, X₁ denotes an actualvelocity of the moving object i (or a velocity of the moving object imeasured with the speedometer having the negligible error), β₀ denotesaccuracy or a systematic error of the speedometer, β₁ denotes themeasurement unit of the speedometer, and εi denotes an error term thatfollows N(0,σ²).
 4. The method of claim 1, wherein the estimating of theparameters comprises setting the independent variable to be bounded toan open interval (−c, c) (where c is any finite value).
 5. The method ofclaim 1, wherein the estimating of the parameters comprises correctingthe independence of the error term to E[ε·f(ϕ)]=0 by setting everymoving object to have same rest mass in the Lorentz velocitytransformation and then applying rapidity ϕ of a moving object measuredat a relativistic center of momentum.
 6. The method of claim 5, whereinthe estimating of the parameters comprises deriving E[ε]=0 andE[ε·sinh(ϕ)]=0 as a population moment condition based on the correctedindependence of the error term, and estimating the parameters of thelinear regression equation using the population regression equation andthe derived population moment condition.
 7. The method of claim 6,wherein the estimating of the parameters comprises estimating theparameters of the linear regression equation after deriving a sampleregression equation and a sample moment condition, which correspond tothe population regression equation and the derived population momentcondition.
 8. The method of claim 7, wherein the estimating of theparameters comprises estimating accuracy and the measurement unit of thespeedometer as the parameters of the linear regression equation throughEquation 2 below: $\begin{matrix}{{{\hat{\beta}}_{0} = {{\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} = {\overset{\_}{Y} - {\frac{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}} \cdot \overset{\_}{X}}}}}{{\hat{\beta}}_{1} = \frac{\sum\limits_{i = 1}^{N}{Y_{i} \cdot {\sinh\left( \phi_{i} \right)}}}{\sum\limits_{i = 1}^{N}{X_{i} \cdot {\sinh\left( \phi_{i} \right)}}}}} & \left\lbrack {{Equation}\mspace{14mu} 2} \right\rbrack\end{matrix}$ where {circumflex over (β)}₀ denotes an estimated accuracyof the speedometer, {circumflex over (β)}₁ denotes an estimatedmeasurement unit of the speedometer,${\overset{¯}{X} = {\frac{1}{N}{\sum_{\overset{`}{l} = 1}^{N}X_{i}}}},{\overset{\_}{Y} = {\frac{1}{N}{\sum_{i = 1}^{N}Y_{i}}}},{\phi_{i} = {\theta_{i} - \theta_{0}}},{\theta_{i} = {\tanh^{- 1}\left( \frac{X_{i}}{c} \right)}},{and}$${\tanh\left( \theta_{0} \right)} = {\frac{\sum_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}.}$9. The method of claim 8, wherein the estimating of the parameterscomprises, based on the estimated parameters of the linear regressionequation, estimating variances and a covariance for the correspondingparameters.
 10. The method of claim 9, wherein the estimating of theparameters comprises, based on the estimated parameters of the linearregression equation, estimating the respective variances for thecorresponding parameters through Equation 3 below: $\begin{matrix}\begin{matrix}{\mspace{79mu}{{{Var}\left( {\hat{\beta}}_{0} \right)} = {{Var}\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1} \cdot \overset{\_}{X}}} \right)}}} \\{= {{{Var}\left( \overset{\_}{Y} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{{Var}\left( {\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {\beta_{0} + {\beta_{1} \cdot X_{i}} + \epsilon_{i}} \right)}} \right)} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {{\frac{1}{N^{2}} \cdot N \cdot \sigma^{2}} + {{\overset{\_}{X}}^{2}{{Var}\left( {\hat{\beta}}_{1} \right)}}}} \\{= {\frac{1}{N}\left( {1 + \frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right){\overset{\_}{X}}^{2}}{2{c^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} \right)\sigma^{2}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 3} \right\rbrack \\{{{Var}\left( {\hat{\beta}}_{1} \right)} = {{\sigma^{2}\frac{N\left( {T - 1} \right)}{2}\frac{S^{2} - C^{2}}{c^{2} \cdot {N^{2}\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}} = {\frac{\left( {S^{2} - C^{2}} \right)\left( {T - 1} \right)}{2{c^{2} \cdot {N\left( {C^{2} - S^{2} - \frac{C}{H}} \right)}^{2}}}\sigma^{2}}}} & \;\end{matrix}$ where Var({circumflex over (β)}₀) denotes a variance for{circumflex over (β)}₀, Var({circumflex over (β)}₁) denotes a variancefor {circumflex over (β)}₁,${S = {\frac{1}{N}{\sum_{i = 1}^{N}{\sinh\left( \theta_{i} \right)}}}},{C = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\left( \theta_{i} \right)}}}}$${T = {\frac{1}{N}{\sum_{i = 1}^{N}{\cosh\left( {2 \cdot \phi_{i}} \right)}}}},{and}$$H = {\frac{N}{\sum_{i = 1}^{N}\frac{1}{\cosh\left( \theta_{i} \right)}}.}$11. The method of claim 9, wherein the estimating of the parameterscomprises, based on the estimated parameters of the linear regressionequation, estimating the covariance between the corresponding parametersthrough Equation 4 below: $\begin{matrix}\begin{matrix}{{{Cov}\left( {{\hat{\beta}}_{0},{\hat{\beta}}_{1}} \right)} = {E\left\lbrack {\left( {{\hat{\beta}}_{0} - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right)\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - {E\left\lbrack {\hat{\beta}}_{0} \right\rbrack}} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - {E\left\lbrack {\hat{\beta}}_{1} \right\rbrack}} \right)} \right\rbrack}} \\{= {E\left\lbrack {\left\{ {\left( {\overset{\_}{Y} - {{\hat{\beta}}_{1}\overset{\_}{X}}} \right) - \left( {\overset{\_}{Y} - {\beta_{1}\overset{\_}{X}}} \right)} \right\}\left( {{\hat{\beta}}_{1} - \beta_{1}} \right)} \right\rbrack}} \\{= {E\left\lbrack {{- \overset{\_}{X}} \cdot \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2}} \right\rbrack}} \\{= {{- \overset{\_}{X}} \cdot {E\left\lbrack \left( {{\hat{\beta}}_{1} - \beta_{1}} \right)^{2} \right\rbrack}}} \\{= {{- \overset{\_}{X}} \cdot {{Var}\left( {\hat{\beta}}_{1} \right)}}}\end{matrix} & \left\lbrack {{Equation}\mspace{14mu} 4} \right\rbrack\end{matrix}$ where Cov({circumflex over (β)}₀,{circumflex over (β)}₁)denotes a covariance between {circumflex over (β)}₀ and {circumflex over(β)}₁.
 12. A parameter estimating system based on correction ofindependence of an error term, the system comprising: a moving objectvelocity acquiring unit configured to acquire a measured velocity, whichis a velocity of a moving object measured with a speedometer beingoperated by an observer and having a specific measurement unit, and anactual velocity of the moving object with respect to the observer; amodeling unit configured to perform modeling of a linear regressionmodel with the measured velocity as a dependent variable and the actualvelocity of the moving object with respect to the observer as anindependent variable; and a parameter estimating unit configured toestimate parameters of a linear regression equation after correcting theindependence of the error term based on Lorentz velocity transformation.13. A computer program stored in a computer-readable recording medium ofclaim 1, the program in which each step of the parameter estimatingmethod based on correction of independence of an error term is performedby a processor of an information processing device.
 14. A computerprogram stored in a computer-readable recording medium of claim 12, theprogram in which each step of the parameter estimating method based oncorrection of independence of an error term is performed by a processorof an information processing device.